Pseudoheterodyne Detection in Optics

Introduction

There is a recent note about pseudo-heterodyne detection in nearfield optics, which is based on measurements I did together with Rolf Brönnimann at the EMPA Dübendorf, Switzerland, on a scanning nearfield optical microscope (SNOM). The detection scheme allows for visualization of optical intensity and phase down to intensities in the lower femtowatt range. First of all, this is two orders of magnitude better than what is possible with an avalanche photodiode module (APD), and furthermore gives you optical phase information. You can achieve higher detection sensitivity by using time-triggered single photon detectors (PMTs or APDs in Geiger mode). These however require a pulsed laser setup.

Please note that the optical phase needs to be preserved with interferometric techniques, because any nowadays available optical detector solely measures the optical intensity. Besides the discussed pseudoheterodyne detection scheme, heterodyne detection and phase shifting interferometry are popular approaches to measure the optical phase.

Difference between Heterodyne and Pseudoheterodyne Detection

Sometimes I get confused reactions on the term “pseudoheterodyne detection”, because “pseudo” sounds like nothing real is going on. To avoid confusions, I’d like to shortly point out how I see the difference between heterodyne and pseudoheterodyne detection in optics.

For both detection techniques, the measurement setup results in interference between the measurement signal and the modulated reference signal. The measurement signal sig and the modulated reference signal ref are both from the same laser source, as shown in the figure below.

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By neglecting losses in the optical setup, the intensity of the interference signal to be detected reads

I\phantom{_{\text{AC}}} = A_{\text{sig}}^2 + A_{\text{ref}}^2 + 2\cdot A_{\text{sig}} \cdot A_{\text{ref}} \cdot \cos(\phi_{\text{MOD}}+\varphi)

which corresponds to the square sum of the corresponding field amplitudes plus the interference term. φ is the optical phase difference between the measurement signal and the reference signal. By keeping the optical phase of the reference signal constant, φ is the optical phase of the sample and therefore the quantity of interest for phase-sensitive measurements. The main idea behind modulation of the reference signal is a spectral separation of the interference term. I therefore introduced an additional modulation ϕ, which accounts for the modulation technique. In total, there are two modulation techniques manipulating exclusively the optical phase of the reference signal:

  1. Frequency-shifting
  2. Phase-modulation

Frequency-shifting

Frequency-shifting can be realized by acousto-optic modulators (AOM). The shift-frequency Δω is in the order of several MHz and a combination of two AOM is often applied for a kHz shift-frequency (1, 2). The interference term for a frequency-shifted reference signal is

I_{\text{AC}} = 2\cdot A_{\text{sig}} \cdot A_{\text{ref}} \cdot \cos(\Delta\omega \cdot t + \varphi)

The detection of φ and the amplitude of the measurement signal is straightforward with a lock-in amplifier that demodulates Δω. Frequency-shifting therefore creates one single interference beat oscillating at Δω. Phase-sensitive detection by frequency-shifting is known as a heterodyne detection technique.

Phase-modulation

Phase-modulation can be realized by electro-optic modulators (EOM) or piezo-electric phase modulators. Depending on the modulation device, the modulation frequency ω has a value from lower kHz up to MHz. The interference term for a phase-modulated reference signal is

I_{\text{AC}} = 2\cdot A_{\text{sig}} \cdot A_{\text{ref}} \cdot \cos(a_{\text{m}}\cdot\cos(\omega_{\text{m}} \cdot t) + \varphi)

where I introduced the angular phase modulation amplitude am, which is also known as the modulation depth or the modulation index. If I want to extract the optical amplitude and phase of the signal, I need to expand the interference signal as a Fourier series. To do so, I first use the cos(a+b) identity to separate the φ terms and then use the expansions for cos(cos) and sin(cos), which introduce lengthy Fourier series. The result is

I_{\text{AC}} = 4 \cdot A_{\text{sig}} \cdot A_{\text{ref}} \cdot \cos(\varphi) \sum\limits_{k=0}^\infty (-1)^k \phantom{_1}J_{2k}(a_{\text{m}})\phantom{_{-}} \cdot \cos[\phantom{(-}2k\phantom{1)}\cdot \omega_{\text{m}} \cdot t] \\ \phantom{I_{\text{AC}}\:}+ 4\cdot A_{\text{sig}} \cdot A_{\text{ref}} \cdot \sin(\varphi) \sum\limits_{k=1}^\infty (-1)^k J_{2k-1}(a_{\text{m}}) \cdot \cos[(2k-1)\cdot \omega_{\text{m}} \cdot t]

The formula above should give you the following two important messages: Firstly, each sideband is a harmonic of the modulation frequency and weighted by the corresponding Bessel function evaluated at the modulation amplitude. Due to this weighting, you observe only a few low order harmonics in practice, whereas high order harmonics vanish. Secondly, the optical in-phase component – the term proportional to cos(φ) on the first line – is on even harmonics, whereas the optical quadrature component – the term proportional to sin(φ) on the second line – is on odd harmonics. As both the optical in-phase and quadrature component are needed for phase-resolved measurements, an even and an odd harmonic of the modulation frequency need to be demodulated by the lock-in amplifier. The most convenient way is to set the modulation amplitude am to 2.63 rad (e.g. 150.7°), where J1(am) = J2(am) = 0.4624, as mentioned in the application note. The first and second harmonic then have the same Bessel coefficient and are demodulated with the lock-in amplifier.

Phase-modulation therefore creates multiple interference beats oscillating at several harmonics of the modulation frequency. Phase-sensitive detection by phase-modulation is known as pseudoheterodyne detection. The main difference to phase-sensitive heterodyne detection is that more than one beat frequency needs to be demodulated in order to extract the optical phase information.

Does Phase-modulation imply Pseudoheterodyne Detection?

Based on this description, one might thinks that optical heterodyne detection is linked to frequency-shifting and pseudoheterodyne detection is linked to phase-modulation. However, this is only true with respect to interferometric phase-sensitive detection. While frequency-shifting creates only a single beat frequency, pseudoheterodyne applications for frequency-shifting do not exist. For phase-modulation on the other hand, there are numerous applications where only the first sideband is demodulated. Accordingly, these techniques are termed heterodyne detection. The outcome of these measurements is not the same as for the pseudoheterodyne detection technique described above. I give a few examples in the following, where I differentiate between interferometric techniques and non-interferometric techniques. For interferometric techniques, the formulas above are also applicable. These techniques measure the first harmonic exclusively, which corresponds to the optical quadrature component discussed above. For non-interferometric techniques, different formalisms are needed to describe oscillating components in the measured intensity.

An example for interferometric heterodyne detection with phase-modulation is interferometric CARS, where optical quadrature component yields the background-free vibrational absorption spectrum (3). Another example for interferometric heterodyne detection with phase modulation is the optical interferometer incorporated into AFMs to measure the cantilever oscillation (4). The cantilever movement modulates the optical phase with a very small modulation amplitude. For both examples, the phase of the reference signal needs to be adjusted for maximum sensitivity, which is when the two interfering signals are in quadrature. A third example is Pound-Drever-Hall (PDH) laser stabilization (5), which creates a feedback signal based on the first phase-modulation sideband.

Another set of examples for heterodyne detection with phase-modulation is found in non-interferometric detection schemes. One example is FM spectroscopy, where the first sideband gives information about absorption and dispersion (6).

While optical heterodyne detection has a counterpart in communication technology (heterodyne receiver with local oscillator), I never heard anyone from the communication community speaking about a pseudoheterodyne receiver. The term pseudoheterodyne detection is so far only used in the context of phase-resolved measurements in optics (7,8).

References

  1. A. Nesci, R. Dändliker, H. Herzig, Quantitative amplitude and phase measurement by use of a heterodyne scanning near-field optical microscope, Opt. Lett. 26, 208-210 (2001)
  2. M. Sandtke, R. J. P. Engelen, H. Schoenmaker, I. Attema, H. Dekker, I. Cerjak, J. P. Korterik, F. B. Segerink, L. Kuipers, Novel instrument for surface plasmon polariton tracking in space and time, Rev. Sci. Instrum. 79, 013704 (2008)
  3. E. Potma, C. Evans, X. Xie, Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging, Opt. Lett. 31, 241-243 (2006)
  4. D. Rugar, H. J. Mamin, R. Erlandsson, J. E. Stern, B. D. Terris, Force microscope using a fiber-optic displacement sensor, Rev. Sci. Instrum. 59, 2337 (1988)
  5. C.G. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, Frequency modulation (FM) spectroscopy, Applied Physics B: Lasers and Optics, Vol. 32, Nr. 3, 145 – 152 (1983)
  6. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, Laser phase and frequency stabilization using an optical resonator, Applied Physics B: Lasers and Optics, Vol. 32, Nr. 3, 97 – 105 (1983)
  7. M. Vaez-Iravani, R. Toledo-Crow, Phase contrast and amplitude pseudoheterodyne interference near field scanning optical microscopy, Appl. Phys. Lett. 62, 1044 (1993)
  8. Nenad Ocelic, Andreas Huber, Rainer Hillenbrand, Pseudoheterodyne detection for background-free near-field spectroscopy, Appl. Phys. Lett. 89, 101124 (2006)